Make sure you've purchased the DLC and didn't accidentally do a donation! If that's the case, please contact Itch Support within 7 days of your purchase, explain your situation, and request a refund.
But if you're 100% certain you bought the DLC correctly, make sure you're signed in to the Itch account you used to make the purchase. Again, you should be able to use the download button on this page (or the download link sent to your email) as long as it's the right account.